#447642

order	512
nilpotency class	3
isoclinism	isoclinic class of #419461
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1050 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,5,4)(3,7,8,6)(9,10,13,19,14,18,28,16)(11,20,22,29,23,15,27,31)(12,24,25,32,26,17,33,35)(21,34,36,39,37,30,38,40);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,21)(13,27)(14,23)(16,31)(17,34)(18,20)(19,29)(22,28)(24,30)(25,38)(26,37)(32,39)(33,36)(35,40);(1,2,5,4)(3,6,8,7)(10,16)(13,28)(15,29)(17,32)(18,19)(20,31)(22,27)(24,35)(25,33)(30,39)(34,40)(36,38);(1,4,5,2)(3,6,8,7)(9,12,14,26)(10,17)(11,21,23,37)(13,25,28,33)(15,30)(16,32)(18,24)(19,35)(20,34)(22,36,27,38)(29,39)(31,40)

Polycyclic presentation:

2^9|a^2=efg,b^2=eg,c^2=eg,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,f^c=fi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 354838);